Theorem spcimdv 2770

Description: Restricted specialization, using implicit substitution. (Contributed by Mario Carneiro, 4-Jan-2017.)

Hypotheses

Ref	Expression
spcimdv.1	|- ( ph -> A e. B )
spcimdv.2	|- ( ( ph /\ x = A ) -> ( ps -> ch ) )

Assertion

Ref	Expression
spcimdv	|- ( ph -> ( A. x ps -> ch ) )

Distinct variable groups:   x, A   ph, x   ch, x

Allowed substitution hints:   ps( x)   B( x)

Proof of Theorem spcimdv

Step	Hyp	Ref	Expression
1		spcimdv.2	. . . 4 |- ( ( ph /\ x = A ) -> ( ps -> ch ) )
2	1	ex 114	. . 3 |- ( ph -> ( x = A -> ( ps -> ch ) ) )
3	2	alrimiv 1846	. 2 |- ( ph -> A. x ( x = A -> ( ps -> ch ) ) )
4		spcimdv.1	. 2 |- ( ph -> A e. B )
5		nfv 1508	. . 3 |- F/ x ch
6		nfcv 2281	. . 3 |- F/_ x A
7	5, 6	spcimgft 2762	. 2 |- ( A. x ( x = A -> ( ps -> ch ) ) -> ( A e. B -> ( A. x ps -> ch ) ) )
8	3, 4, 7	sylc 62	1 |- ( ph -> ( A. x ps -> ch ) )

Syntax hints:   -> wi 4   /\ wa 103   A.wal 1329   = wceq 1331   e. wcel 1480

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688

This theorem is referenced by:  spcdv  2771  rspcimdv  2790